Is pooling a kind of dropout?
Dropout and Max-pooling are performed for different reasons.

Dropout is a regularization technique, which affects only the training process (during evaluation, it is not active). The goal of dropout is reduce unnecessary feature dependencies in the network, allowing it to be simpler and improves its generalization abilities (reduces overfitting). In simple terms, it helps the model to learn that some features are an "OR" and not an "AND" requirements.

Max-pooling is not a regularization technique and it is part of the model's architecture, so it is also used during evaluation. The goal of max-pooling is to down-sample an input representation. As a result the model becomes less sensitive to some translations (improving translation invariance).

As for your last question, yes. dropout can be used after a pooling layer.